How well-behaved are higher-order perturbation solutions?

They are not well-behaved. The main problem is that one cannot control the radius of convergence when using perturbation techniques. Just outside the radius of convergence, higher-order approximations can easily behave extremely badly, and even within the radius of convergence one can expect higher- but finite-order perturbation solutions to display problematic oscillations. In contrast, with projection methods one can control the radius of convergence. Pruning, the solution proposed to deal with explosive behavior of higher-order perturbation solutions, is shown to be highly distortionary. A simple alternative based on short samples and rejection sampling is proposed and shown to be much less distortive.